Title: Parameterized Wasserstein Hamiltonian Flow Show Chinese title

Title: 参数化Wasserstein哈密顿流

Abstract: In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use parameterized function as push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish error analysis results for the continuous time approximation scheme in Wasserstein metric. For the numerical implementation, we use neural networks as push-forward maps. We apply an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as total energy. The computation is done by fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid the error introduced by stochastic gradient descent (SGD) methods, which is usually hard to quantify and measure. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics. Show Chinese abstract

Abstract: 在这项工作中，我们提出了一种数值方法来计算Wasserstein哈密顿流（WHF），这是一种在概率密度流形上的哈密顿系统。许多著名的偏微分方程系统可以重新表述为WHF。我们使用参数化函数作为推进映射来表征WHF的解，并将偏微分方程转换为有限维常微分方程系统，该系统是参数流形相空间中的哈密顿系统。我们在Wasserstein度量下建立了连续时间逼近格式的误差分析结果。对于数值实现，我们使用神经网络作为推进映射。我们应用一种有效的辛格式求解推导出的哈密顿常微分方程系统，从而使方法保留了一些重要的守恒量，例如总能量。计算完全由确定性的辛积分器完成，而无需任何神经网络训练。因此，我们的方法不涉及网络参数的直接优化，从而避免了随机梯度下降（SGD）方法引入的误差，这种误差通常难以量化和测量。所提出的算法是一种基于抽样的方法，能够很好地扩展到更高维问题。此外，该方法还通过参数化常微分方程动力学提供了原始WHF的拉格朗日和欧拉视角之间的另一种联系。